On parametric types of Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials via Golden calculus

: This paper aims to give generating functions for the new family of polynomials, which are called parametric types of the Apostol Bernoulli-Fibonacci, the Apostol Euler-Fibonacci


Introduction
Special polynomials, generating functions, and the trigonometric functions are used not only in mathematics but also in many branches of science such as statistics, mathematical physics, and engineering.
Let N, Z, R and C indicate the set of positive integers, the set of integers, the set of real numbers, and the set of complex numbers, respectively. Let α ∈ N 0 = N ∪ {0} and λ ∈ C (or R).
(1.12) Remark 1.1. Note that the symbols c and s occurring in the superscripts on the left-hand sides of these last Eqs (1.7)-(1.12) indicate the presence of the trigonometric cosine and the trigonometric sine functions, respectively, in the generating functions on the corresponding right-hand sides.
The motivation of this paper is to obtain F-analogues of the Eqs (1.7)-(1.12) with the help of the Golden calculus. Namely, we define the parametric Apostol Bernoulli-Fibonacci, the Apostol Euler-Fibonacci, and the Apostol Genocchi-Fibonacci polynomials by means of the Golden Calculus. Utilizing the Golden-Euler formula and these generating functions with their functional equations, numerous properties of these polynomials are given. The special cases of these polynomials and numbers are studied in detail. The rest of this paper is structured as follows. In Section 2, we present some key definitions and properties that are crucial to Golden calculus. Then, with the help of the Golden calculus, we mention some polynomials that have been previously defined in the literature. In Section 3, considering the properties of Golden calculus, we introduce six families of two-parameter polynomials with the help of Golden trigonometric functions and exponential functions. Then, in the three subsections of this section, we examine the various properties of these polynomials defined with the help of generating functions and their functional equations.

Golden calculus
In this part of the our paper, we mention some definitions and properties related to Golden calculus (or F-calculus).
The Fibonacci sequence is defined by means of the following recurrence relation: where F 0 = 0, F 1 = 1. Fibonacci numbers can be expressed explicitly as where α = 1+ 2 . α ≈ 1, 6180339 . . . is called Golden ratio. The golden ratio is frequently used in many branches of science as well as mathematics. Interestingly, this mysterious number also appears in architecture and art. Miscellaneous properties of Golden calculus have been defined and studied in detail by Pashaev and Nalci [21]. Therefore, [21] is the key reference for Golden calculus. In addition readers can also refer to Pashaev [22], Krot [23], and Ozvatan [24].
The product of Fibonacci numbers, called F-factorial was defined as follows: where F 0 ! = 1. The binomial theorem for the F-analogues (or-Golden binomial theorem) are given by in terms of the Golden binomial coefficients, called as Fibonomials n k F = F n ! F n−k !F k ! with n and k being nonnegative integers, n ≥ k. Golden binomial coefficients (or-Fibonomial coefficients) satisfy the following identities as follows: The Golden derivative defined as follows: 3) The Golden Leibnitz rule and the Golden derivative of the quotient of f (x) and g(x) can be given as respectively. The first and second type of Golden exponential functions are defined as and Briefly, we use the following notations throughout the paper and Using the Eqs (2.2), (2.4), and (2.5), the following equation can be given The Fibonacci cosine and sine (Golden trigonometric functions) are defined by the power series as and For arbitrary number k, Golden derivatives of e kx F , E kx F , cos F (kx) , and sin F (kx) functions are and (2.14) Using (2.4), Pashaev and Ozvatan [25] defined the Bernoulli-Fibonacci polynomials and related numbers. After that Kus et al. [26] introduced the Euler-Fibonacci numbers and polynomials. Moreover they gave some identities and matrix representations for Bernoulli-Fibonacci polynomials and Euler-Fibonacci polynomials. Very recently, Tuglu and Ercan [27] (also, [28]) defined the generalized Bernoulli-Fibonacci polynomials and generalized Euler-Fibonacci polynomials, namely, they studied the Apostol Bernoulli-Fibonacci and Apostol Euler-Fibonacci of order α as follows:

Generalized parametric types of Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials via Golden calculus
Krot [23] defined the fibonomial convolution of two sequences as follows. Let a n and b n are two sequences with the following generating functions then their fibonomial convolution is defined as c n = a n * b n = n l=0 n k F a l b n−l .
So, the generating function takes the form Let p, q ∈ R. The Taylor series of the functions e pt F cos F (qt) and e pt F sin F (qt) can be express as follows: By virtue of above definitions of C n,F (p, q) and S n,F (p, q) and the numbers B (α) n,F (λ) , E (α) n,F (λ) and G (α) n,F (λ), we can define two parametric types of the Apostol Bernoulli-Fibonacci polynomials, the Apostol Euler-Fibonacci polynomials, and the Apostol Genocchi-Fibonacci polynomials of order α, as follows: n,F (p, q; λ) = G (α) n,F (λ) * S n,F (p, q) , whose exponential generating functions are given, respectively, by respectively. If we take p = 0 in (3.11)-(3.13), we obtain Apostol Bernoulli-Fibonacci numbers B n,F (λ), Apostol Euler-Fibonacci numbers E n,F (λ), and Apostol Genocchi-Fibonacci numbers G n,F (λ) .

14)
and Proof. By applying (3.5), we first derive the following functional equation: Comparing the coefficients of t n on both sides of this last equation, we have which proves the result (3.14). The assertion (3.15) can be proved similarly. □ Remark 3.5. We claim that Theorem 3.2. For every n ∈ N, following identities hold true:

18)
and Proof. Using (3.5) and applying the Golden derivative operator ∂ F ∂ F p , we obtain By comparing the coefficients of t n on both sides of this last equation, we arrive at the desired result (3.16). To prove (3.18), using (3.5) and applying the Golden derivative operator ∂ F ∂ F q , we find that Comparing the coefficients of t n on both sides of this last equation, we arrive at the desired result (3.18  Proof. Setting α = 1 in (3.5) and using (3.1), we find that Comparing the coefficients of t n on both sides of this last equation, we arrive at the desired result (3.20). Equation (3.21) can be similarly derived. □ Theorem 3.4. The following identities hold true:

22)
and Proof. Setting α = 1 in (3.5) and using (2.9), we find that Comparing the coefficients of t n on both sides of this last equation, we arrive at the desired result (3.22). Equation (3.23) can be similarly derived. □ Theorem 3.5. The following identities hold true: Proof. Using the following equation for the proof of (3.24), we have Considering B (c,1) 0,F (p, q; λ) = 0, and doing some calculations, we arrive at the desired result (3.24). Equation (3.25) can be similarly derived. □ Theorem 3.6. The following identities hold true:
Comparing the coefficients of t n on both sides of this last equation, we arrive at the desired result (3.26). Equation (3.27) can be similarly derived. □ Theorem 3.7. Determinantal forms of the cosine and sine Apostol Bernoulli-Fibonacci polynomials are given by Proof. Equation (3.24) cause the system of unknown (n + 2)-equations with B (c,1) n,F (p, q; λ) , (n = 0, 1, 2, . . .) . Then we apply the Cramer's rule to solve this equation. We obtain the desired result. In a similar way, we can obtain the determinantal form for sine Apostol Bernoulli-Fibonacci polynomials. □ In subsections 3.2 and 3.3, we give the some basic properties of the polynomials E (c,α) n,F (p, q; λ) , E (s,α) n,F (p, q; λ) , G (c,α) n,F (p, q; λ) , and G (s,α) n,F (p, q; λ) . Their proofs run parallel to those of the results presented in this subsection; so, the proofs are omitted.

Conclusions
Our aim in this article is to define the F-analogues of the parametric types of the Apostol Bernoulli, the Apostol Euler, and the Apostol Genocchi polynomials studied by Srivastava et al. [15,19]. Namely, we have defined parametric types of the Apostol Bernoulli-Fibonacci, the Apostol Euler-Fibonacci, and the Apostol Genocchi-Fibonacci polynomials using the Golden calculus and investigated their properties. In our future work, we plan to define the parametric types of some special polynomials with the help of Golden calculus and to obtain many combinatorial identities with the help of their generating functions.